I am a researcher in medical informatics, and statistics is a huge part of my job, though I am not a classically-trained statistician.
First, I would like to offer a stark contrast between two types of statisician: 1) statisticians of the old mold who are wedded to SAS and related tools and 2) research statisticians who employ modern methods such as Bayesian statistics and rather advanced calculus. The former tend to mold all problems into what is available in the canon of SAS routines, while the latter are capable of creating custom models that suit the problem at hand.
Then, there is a new breed of scientist -- the data scientist -- who tends to use black-box machine learning methods and the classical techniques, as programs such as SAS and R have "democratized" the field. I agree with the common gripe of many traditionally-trained statisticians who object that these "data scientist" tend not to understand the statistical background of these computer codes. In fact, it is easy to download R onto one's computer and start firing data through, with little regard for the merits of the model or its results. (Not all data scientists are like this, but I'm simply stating a general observation.)
Another problem with statistics is that it can be very confusing, understanding just what things like p-values mean. After a first course in statistics, it leaves many with a bad taste -- either being terribly confusing, or rather boring. In my opinion, this is because of traditional (frequentist) statistics, which have their origins from luminaries such as Fisher and Pearson.
The "action" today is in Bayesian statistics. This formulation allows for statistical concepts to be expressed is ways that (I believe) most people can understand. But executing Bayesian statistics mandates that one understand the underlying formulation of models; in general, they are not black-box methods. Furthermore, they can be quite computationally-expensive for large data.
Statistics is suffering from perceptions of being a button-pushing, boring profession. As has happened in many other fields (e.g. computational chemistry and CFD), computer programs have democratized the field so that those who have not had years of dedicated study and training can execute statistical models. In my experience, this can be a good thing, or a very bad thing. Another issue is that there is a significant build-up of half a century of code and protocols in both industry (think big business analysis) and government agencies (think FDA).
But modern statistics is actually a hot field. Provided that one understands the background, and is willing to go the extra mile to write custom code, the rewards are endless.